On non-principal arithmetical numberings and families

The paper studies \(\Sigma^0_n\) -computable families ( \(n\geqslant 2\) ) and their numberings. It is proved that any non-trivial $\Sigma^0_n$-computable family has a complete with respect to any of its elements $\Sigma^0_n$-computable non-principal numbering. It is established that if a $\Sigma^0_n$-computable family is not principal, then any of its $\Sigma^0_n$-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal $\Sigma^0_n$-computable numberings. It is also shown that for any $\Sigma^0_n$-computable numbering \(\nu\) of a $\Sigma^0_n$-computable non-principal family there exists its $\Sigma^0_n$-computable numbering that is incomparable with $\nu$. If a non-trivial $\Sigma^0_n$-computable family contains the least and greatest elements under inclusion,then for any of its $\Sigma^0_n$-computable non-principal non-least numberings $\nu$ there exists a $\Sigma^0_n$-computable numbering of the family incomparable with $\nu$. In particular, this is true for the family of all $\Sigma^0_n$-sets and for the families consisting of two inclusion-comparable $\Sigma^0_n$-sets (semilattices of the $\Sigma^0_n$-computable numberings of such families are isomorphic to the semilattice of \(m\) -degrees of $\Sigma^0_n$-sets). MSC Classification: 03D45